THE 2-GENERATORS FOR CERTAIN SIMPLE PERMUTATION GROUPS OF SMALL DEGREE

THE 2-GENERATORS FOR CERTAIN SIMPLE

PERMUTATION GROUPS OF SMALL DEGREE

Osamu Higuchi and Izumi Miyamoto

(Received February 20, 1998)

Abstract. The 2-generators ofA7,A8,A9,M11,M12andM22 are computed

up to equivalence of their automorphism groups and the automorphism group of a free group of rank 2. The actions of the automorphism group of the free group on their dening subgroups are also computed.

AMS 1991 Mathematics Subject Classi
cation. Primary 20F05 Secondary 20B99.

Key words and phrases. Generator and relation, permutation group, simple group.

x

1. Introduction

Let a group

G

be generated by two ordered pairs (

x

1

x

2) and (

y

1

y

2). Then

these satisfy the same dening relations for

G

if and only if there exists an element



in the automorphism group

Aut

(

G

) so that (

x

1

x

2) = (

y

1

y

2).

Suppose (

y

1

y

2) = (

x

1

x

1

x

2) or (

y

1

y

2) = (

x

2

x

1), then substituting

x

1 =

y

1

x

2 =

y

;1

1

y

2

or

x

1 =

y

2

x

2 =

y

1

in the relations with respect to (

x

1

x

2)

respectively, we have new relations for

G

with respect to (

y

1

y

2). Let

V

(

G

)

be the set of all generating pairs (

x

1

x

2) of

G

and let b

V

(

G

) be the set of the orbits of

Aut

(

G

) on

V

(

G

). Then the transformations (

x

1

x

2)

! (

x

1

x

1

x

2)

and (

x

1

x

2) ! (

x

2

x

1) induce permutations on

V

(

G

) and they also induce

permutations on

V

b(

G

), since every orbit of

Aut

(

G

) on

V

(

G

) is characterized

as the subset of

V

(

G

) having same dening relations. In the present paper for

G

=

A

7

A

8

A

9

M

11

M

12 and

M

22, one of the alternating groups of degree

7, 8, 9 or the Mathieu groups of degree 11, 12 and 22, we compute

V

b(

G

) and

next compute how the permutation group generated by the transformations acts on each of its orbits on

V

b(

G

). In 4] and 6] the same was acheived for

A

5

and in 10] for

A

6 and

PSL

(2

7). Some related results for

A

6 and

A

7 are seen

in 7] and 8]. The authors note with thanks that computations were carried out with the group algorithm programming system GAP9].

Let

F

be a free group generated by

u

and

v

. A normal subgroup

N

of

F

is said to be a

G

-dening subgroup if

G

=

F=N

. Then the elements of

Aut

(

F

)

given by (

u
v

) ! (

u
u

;1

v

) and (

u
v

)

! (

v
u

) generate an equivalent

permutation group on the set of

G

-dening subgroups of

F

as a permutation group on

V

b(

G

), by a result of 6]. If

G

is a nite non-abelian simple group, the

action of the automorphism group of a free group is likely to be a symmetric or an alternatng group on each of its orbits on the

G

-dening subgroups in most cases, and it is intended to prove this fact under some general conditions in some references (cf. 2], 3]). For the above groups we computed how a free group with 2 generators acts on the

G

-dening subgroups. The results are shown in tables at the end of the present paper and the summary is as follows.

Theorem 1.1.

Let

F

be a free group generated by two elements and suppose

G

=

A

7,

A

8,

A

9,

M

11,

M

12 or

M

22. Then the action of

Aut

(

F

) on each of its

orbits on the

G

-dening subgroups of

F

is one of the types

S

m,

A

m, 2o

S

m,

2m;1:

S

m, 2m;1

A

m,

S

mo

S

3,

A

m o

S

3,

A

3 m

S

3,

A

3 m

S

y 3,

A

3 m2

S

3,

A

3 m22

S

3,

A

3 m22

S

y 3,

S

m2,

A

m2, 2m;1:

S

m2, 2m;1

A

m2,

A

3 m22

S

32 and

A

3 m22

S

y 32. In particular

m

is

even if the action has a normal 2-subgroup.

Here we explain some of the notation in Theorem 1.1. First 2o

S

n is

a wreath product of a cyclic group of order 2 by a symmetric group

S

n of degree

n

and, as a permutation group, is of degree 2

n

with

n

blocks of length 2 and quotient action as

S

n. 2n;1:

S

n, a non-split extension of 2n;1 by

S

n, is a subgroup of 2o

S

n with the kernel consisting of all products of an even

number of 2-cycles on the blocks. Similarly 2n;1

A

n, a semidirect product of 2n;1 by

A

n, is also a subgroup of 2o

S

n. Next 2n ;1

A

n2 is of degree two times that of 2n;1

A

n, so of degree 4

n

, and the right-most symbol 2 means that a cyclic group of order 2 interchanges the two orbits of 2n;1

A

n, which is in fact induced by the transformation (

x

1

x

2)

!(

x

2

x

1). Also between two families

of wreath products

A

no

S

3 and

S

n o

S

3of degree 3

n

(with 3 blocks of length

n

)

there exist two families of incomplete wreath products which are denoted by

A

3

n2

S

3 and

A

3

n22

S

3 respectively, and nally there are two similar subgroups of

S

no

S

3 which are denoted by

A

3 n

S

y 3 and

A

3 n22

S

y

3. In these cases there are no

subgroups isomorphic to

S

3 whose transpositions centralize one component of

A

3

n and interchange the remaining two components.

As

PSL

(2

q

) is a permutation group of degree

q

+ 1, our program may also work for

PSL

(2

q

) when

q

is a small prime power. But we may be able to treat it as a matrix group with nite eld elements inGAP because it will

save memory space for storing

V

b(

G

). So in the present paper we restrict our

attension to above permutation groups of small degree.

x

2. Computation and observations

Aut

(

F

) is generated by Nielsen transformations (see e.g. 5] chapter 3). They permute the

G

-dening subgroups of

F

and the action is equivalent to the following permutations on

V

(

G

) by 6]: (i) (

x
y

) ! (

x
x

;1

y

) (ii) (

x
y

) !

(

x
yx

;1) (iii) (

x
y

)

! (

y
x

) (iv) (

x
y

) ! (

x
y

;1). The permutations (i)

and (ii) are conjugate under

Inn

(

G

), so they act on

V

b(

G

) as a same

permuta-tion. Let the permutations (i) and (iii) be denoted by



;1 and



, respectively.

Then



is (

x
y

) ! (

x
xy

). The permutation (iv) is obtained as a conjugate

of



;1



. Hence the permutation group on b

V

(

G

) induced by

Aut

(

F

) is generated by the action of



and



on

V

b(

G

). In this way the action of

Aut

(

F

)

on

V

b(

G

) is obtained.

The automorphism groups of alternating groups are symmetric groups, ex-cept in the case

A

6. So in our cases

Aut

(

G

)

 =

S

nif

G

=

A

n.

Aut

(

M

11)  =

M

11, but

Aut

(

M

12) 

=

M

122 and we construct it as a permutation group of degree

24 interchanging two orbits of

M

12. Finally

Aut

(

M

22) 

=

M

222 is of degree 22

(see e.g. 1]).

We compute

Aut

(

G

)-conjugacy classes of pairs (

x

1

x

2) with

x

1

x

2 2

G

and enumerate the representatives of the conjugacy classes such that

x

1 and

x

2 generate

G

, which are b

V

(

G

). Next for each pair (

x

1

x

2) in b

V

(

G

) we see what elements in

V

b(

G

) are

Aut

(

G

)-conjugate to (

x

1

x

1

x

2) and (

x

2

x

1)

respectively. This determines the action of



and



on

V

b(

G

). We list the

representatives and lengths of the orbits of the permutation group on

V

b(

G

)

induced by

Aut

(

F

). Finally we will determine how this permutation group acts on each of its orbits.

Consider any one of the orbits of

Aut

(

F

) on

V

b(

G

) and let it be denoted by

. Let

a

and

b

be permutations obtained by restricting the action of



and



to  respectively and let

H

= h

a
b

i. We ask whether

H

is primitive by

using theGAP-commandIsPrimitive. If it is, then we seek an element of

H

satisfying the following.

Theorem 2.1.

(Jordan, see 11]) If a primitive group of degree

n

contains a cycle of prime degree

p

with

n

;

p

3, then the group is either alternating

or symmetric.

Then computing the signs of permutation

a

and

b

, we can determine whether

Here we note that we may use the MAGMA-command IsAlternatingor IsSymmetric. However it was sucient to use the above theorem to determine alternating or symmetric in theGAP-system.

If

H

is not primitive, then the GAP-commandBlocks gives blocks of

im-primitivity and in our cases the length of a block was 2 or one third of the length of . Let  be the set of the blocks, let

H

,

a

and

b

be the images of

H

,

a

and

b

on  respectively, and let

K

be the kernel of

H

!

H

. In either

case we compute

a

and

b

and if the degree of

H

is rather large, we apply the above method in order to see if

H

is alternating or symmetric, and if not, we compute it directly. Then we use the following lemmas.

Lemma 2.2.

Let jj = 2

m

and suppose that

H

has a block of length 2

and that

A

m 

H

. If there exists an element

h

2

H

such that its order j

h

j

is dierent from j

h

j and that

h

has a xed point on , then

K

contains all

products of an even number of 2-cycles on blocks, hence 2m;1

j

K

j, and

H

contains a subgroup isomorphic to

A

m. Furthermore if

H

contains an element

k

with

sign

(

k

) =;1 and

sign

(

k

) = 1, then

H



= 2o

H

.

Proof. The condition on

h

implies that

K

6= 1. Then we have

K

contains all

products of an even number of 2-cycles on blocks, since

A

m 

H

and since the

length of the block is 2. Let the

j

-th block be

j

=f

j
j

0

g. If we take a 3-cycle

s

and an odd-cycle

t

of

H

with

s

= (1

2

3) and

t

= (3

4

::

2

l

+ 1), then we may assume that

s

is a product of two 3-cycles and so is

t

and we may setf

j
j

0

gso

that

s

= (1

2

3)(10

20

30) and

t

= (3

4

::

2

l

+ 1)(30

40

::

(2

l

+ 1)0). Hence we

haveh

s
t

i 

=

A

2l+1. If

m

= 2

l

+2, then we may have either

r

1 = (2

l

2

l

+1

2

l

+

2)((2

l

)0

(2

l

+ 1)0

(2

l

+ 2)0) or

r

2 = (2

l

(2

l

+ 1)

0

2

l

+ 2)((2

l

)0

2

l

+ 1

(2

l

+ 2)0).

In the latter case

r

2

w

=

r

1 with

w

= (2

l

+ 1

(2

l

+ 1)

0)(2

l

+ 2

(2

l

+ 2)0) 2

K

.

So in either case we haveh

s
t
r

1

i 

=

A

2l+2 as a subgroup of

H

.

The condition on

k

implies that there exists

k

0 in the above alternating

subgroup with

k

=

k

0 and that

k

;1

k

0 is a product of an odd number of

trans-positions. Hence

K

contains a transposition (

j
j

0) for every

j

. If

H

contains

r

= (1

2), then we may assume

r

= (1

2

10

20), (1

20

10

2), (1

20)(10

2) or

(1

2)(10

20). The last case clearly gives a subgroup isomorphic to

S

m. For the remaining cases,

r

(1

10),

r

(2

20) and (1

10)

r

(1

10) are equal to (1

2)(10

20)

respectively. Thus the last assertion holds. 2

The existence of

k

in Lemma 2.2 depends only on the signs of

a

,

b

,

a

and

b

as

sign

(

k

) =;1 and

sign

(

k

) = 1. We note that the case for 2m ;1:

S

m occurs when an odd permutation of

H

is also odd in

H

in Theorem 1.1.

If

H

has 3 blocks, we have

H

=

S

3. In this case we compute some stabilizing

whether one component of the stabilizer is primitive and so on. Thus as above we nd whether

K

acts on a block as an alternating or a symmetric group. Next we nd an element not acting with the same order on all blocks. This gives

A

3

m 

K

, where jj= 3

m

. Then computing the signs of the stabilizing

elements on each block of , we determie

K

to be one of

A

3

m,

A

3 m2,

A

3 m22 or

S

3 m.

Lemma 2.3.

If

K

=

A

3 m or

A

3

m22 and if any transposition of

H

is an odd

or even permutation on  according as

m

is even or odd, then

H

does not contain a subgroup isomorphic to a split extension of

K

by

S

3. Otherwise

H

contains a subgroup isomorphic to such a split extension.

Proof. Let  1 = f1

2

m

g,  2 = f1 0

20

m

0 g and  3 = f1"

2"

m

"g be the blocks and let

t

be a transposition of

H

interchanging 

1 and 2.

If

t

itself is an involution and xes every point of 3, then

t

does not satisfy the

condition of the Lemma. In this case

t

and one of its conjugate interchanging 2and 3generate a subgroup isomorphic to

S

3, any of whose involutions

cen-tralizes one component of

K

. Then this subgroup and

K

generate a subgroup of

H

isomorphic to a split extension of

K

by

S

3.

Let

t

2 be the constituent of

t

2on 

2 and let

t

3be the constituent of

t

on 3.

Since

A

3

m

K

, by taking a product of

t

and an element of

K

, we can choose

t

2

either an identity or a transposition, and so does

t

3. By the above argument we

may assume that both of

t

2 and

t

3 are not identities. If

t

2 is a transposition,

then we may set

t

= (1

10

2

20)

and

t

2 = (1

2)(10

20). Then we have

A

3 m22 

K

, since

t

2

2

K

. By taking a product of

t

and (1

0

20)(100

200) which is

a conjugate of

t

2, we may assume for the Lemma that

t

2is an identity and

t

3 is

a transposition. Hence there exist an element (1

10)(2

20)

(

m
m

0)(100

200).

Then this element together with its conjugate (1

2)(10

100)(20

200) (

m

0

m

00)

generates a subgroup isomorphic to

S

3 but its involution does not centalizes a

component of

K

. If

K

=

A

3

m2, then (1

2)(10

20)(100

200)

2

K

, and the product

of this element and the last involution becomes an involution interchanging 2 and 3 and stabilizing all the points of 1. So in this case as in the rst

paragraph we have a subgroup of

H

isomorphic to a split extension of

K

by

S

3. Now it is easy to see that these two types of subgroups isomorphic to

S

3

are distinguished by the signs in the Lemma. 2

In fact we rst see whether the subgrouph

a
a

biof

H

is transitive on . If

it is transitive, then the method mentioned above gives the structure of

H

. If not, we see that it has two orbits on  which are interchanged by

b

and how the subgroup acts on one of its orbits by the above argument. This completes the computation.

Proposition 2.4.

If an element (

x
y

) in

V

(

G

) is not conjugate to (

x

;1

y

;1)

in

Aut

(

G

) then they make a block of length 2 in an orbit of

Aut

(

F

) on

V

b(

G

) Proof. First we note that the transformation (

u
v

) ! (

u

;1

v

;1) is

con-tained in

Aut

(

F

). So such a mutually inverse pair of elements of

V

(

G

) as in Proposition 2.4 represents a pair of elements in a same orbit of

Aut

(

F

) on

b

V

(

G

). Now (

x
y

) goes to (

x
xy

) and (

x

;1

y

;1) goes to (

x

;1

x

;1

y

;1) under

a

. The latter is conjugate to (

x

;1

(

xy

);1) by

x

;1. Thus a mutually inverse

pair of elements in

V

(

G

) goes to another such a pair under

a modulo Aut

(

G

). Clearly the same statement holds for

b

. Hence such a pair makes a block of an orbit of

Aut

(

F

) on

V

b(

G

).

2

In all cases in Theorem 1.1 that have blocks of length 2 the blocks are in fact made up of these mutually inverse pairs. In the remaining cases (

x
y

) is conjugate to (

x

;1

y

;1) in

Aut

(

G

), in which case they represent a same

element of

V

b(

G

).

TABLES

In tables below 'degree' column shows the lengths of the orbits of

Aut

(

F

) on

b

V

(

G

) and '

Aut

(

F

)' column gives the actions of

Aut

(

F

) on its orbits on

V

b(

G

).

The 2-generators are representatives

modulo Aut

(

G

) of the orbits of

Aut

(

F

) on

V

b(

G

). In cases for

M

11,

M

12 and

M

22, the 2-generators are given as

prod-ucts of specic generators

a

and

b

below each table.

No. 2-generators of

A

7 degree

Aut

(

F

)

1 (1

2)(3

4), (1

3

5

2

4

6

7) 16

A

16 2 (1

2

3)(4

5)(6

7), (2

3

4)(5

6

7) 21

A

21 3 (1

2

3)(4

5)(6

7), (2

4

3

6

7) 21

S

21 4 (1

2)(3

4), (2

5

3

6)(4

7) 24

A

3 82

S

3 5 (1

2)(3

4), (2

3)(4

5

6

7) 30

S

10 o

S

3 6 (1

2)(3

4), (1

3

2

4

5

6

7) 36

S

36 7 (1

2)(3

4), (1

3)(2

5)(4

6

7) 36

A

3 122

S

3 8 (1

2)(3

4), (1

3

6

4

7

2

5) 36

A

36 9 (1

2)(3

4), (2

3

5

6

7) 40

S

40 10 (1

2

3)(4

5

6), (2

3

4

6

7) 48 223

A

24 11 (1

2

3

4)(5

6), (2

3)(4

5

7

6) 56 227:

S

28 12 (1

2)(3

4), (2

5

4

6

7) 72

A

72 13 (1

2

3)(4

5)(6

7), (2

3

4

5

6) 84 2o

S

42 14 (1

2

3)(4

5)(6

7), (2

4)(3

6

5

7) 84 2o

S

42 15 (1

2

3), (2

4)(3

5

6

7) 120 259

A

60 16 (1

2

3)(4

5)(6

7), (2

4)(3

5

6

7) 192 247

A

482

No. 2-generators of

A

8 degree

Aut

(

F

) 1 (1

2

3)(4

5

6), (1

4)(2

6

8

3

5

7) 15

S

5 o

S

3 2 (1

2

3)(4

5)(6

7), (3

4)(5

6

8

7) 42

A

3 142

S

3 3 (1

2

3)(4

5)(6

7), (1

4)(2

6

5

7

3

8) 45

S

15 o

S

3 4 (1

2)(3

4), (2

5

7

3

4

6

8) 90

S

90 5 (1

2)(3

4), (1

3

5)(2

4

6

7

8) 96

S

96 6 (1

2)(3

4)(5

6)(7

8), (2

3)(4

5)(6

7

8) 96

A

32 o

S

3 7 (1

2)(3

4), (1

3

2

5

7)(4

6

8) 198

S

198 8 (1

2)(3

4), (2

5

3

6

8

4

7) 252

A

252 9 (1

2)(3

4), (2

3

4

5

6

7

8) 260 2o

S

130 10 (1

2)(3

4), (2

3

5

6

7

8

4) 270

S

270 11 (1

2)(3

4)(5

6)(7

8), (2

3

4

5

6

7

8) 384

S

384 12 (1

2)(3

4), (2

5

3

6

4

7

8) 432 2215:

S

216 13 (1

2

3)(4

5)(6

7), (2

3

4

5

6

8

7) 480 2239

A

240 14 (1

2

3)(4

5)(6

7), (2

3

4)(5

6

8) 540 2o

S

270 15 (1

2)(3

4), (2

3

5

7

4

6

8) 768 2383

A

384 16 (1

2

3)(4

5)(6

7), (3

4)(5

6

7

8) 1092 2o

S

546 17 (1

2

3)(4

5)(6

7), (3

4

5

6)(7

8) 1092 2o

S

546 18 (1

2

3)(4

5)(6

7), (2

3

4

5

6

7

8) 1296 2647:

S

648

No. 2-generators of

A

9 degree

Aut

(

F

)

1 (1

2)(3

4)(5

6)(7

8), (1

3

2

4

5

7

9

6

8) 25

A

25 2 (1

2

3)(4

5)(6

7), (3

8

4

6

5

7

9) 36

A

36 3 (1

2)(3

4), (1

3

5

7

2

4

6

8

9) 72

A

72 4 (1

2)(3

4), (1

3

2

4

5

6

7

8

9) 81

S

81 5 (1

2)(3

4), (2

3

5

6

7

8

9) 90

S

90 6 (1

2)(3

4), (1

5)(2

6

3

7

9)(4

8) 114

S

114 7 (1

2)(3

4), (2

3)(4

5

6

7

8

9) 135

A

3 452

S

3 8 (1

2)(3

4), (1

5)(2

6

3

7)(4

8

9) 138

A

46 o

S

3 9 (1

2)(3

4), (1

3

6

4

7

8

9

2

5) 162

S

162 10 (1

2)(3

4), (2

5

3

6)(4

7

8

9) 222

A

3 74

S

y 3 11 (1

2)(3

4)(5

6)(7

8), (2

3)(4

5

7

9

6

8) 231

A

3 772

S

3 12 (1

2)(3

4)(5

6)(7

8), (2

3)(4

5

7

9

8

6) 243

A

3 81

S

y 3 13 (1

2

3)(4

5)(6

7), (3

4

5

8

9

7

6) 252

A

252 14 (1

2

3)(4

5)(6

7), (3

4)(5

8

7

9) 300

A

3 1002

S

3 15 (1

2

3)(4

5)(6

7), (3

4

6)(5

8)(7

9) 315

S

315 16 (1

2)(3

4), (2

5

4

6

7

8

9) 324

S

324 17 (1

2)(3

4), (1

3)(2

5)(4

6

7

8

9) 360

A

3 1202

S

3 18 (1

2)(3

4)(5

6)(7

8), (2

3)(4

5

9)(6

7) 432

S

432 19 (1

2)(3

4)(5

6)(7

8), (2

3

5)(4

7

9

8

6) 460

A

460 20 (1

2)(3

4)(5

6)(7

8), (2

3

5

7

4)(6

9

8) 486

S

486

No. 2-generators of

A

9 (continued) degree

Aut

(

F

) 21 (1

2)(3

4)(5

6)(7

8), (2

3)(4

5

6

9

8

7) 567

A

3 1892 2

S

y 3 22 (1

2)(3

4)(5

6)(7

8), (2

3

5

7

4)(6

8

9) 574

S

574 23 (1

2)(3

4)(5

6)(7

8), (2

3

4

5

7)(6

9

8) 612 2o

S

306 24 (1

2)(3

4), (1

3

2

5

4

6

7

8

9) 828 2o

S

414 25 (1

2

3)(4

5)(6

7), (2

3

4)(5

6

8

9

7) 864

A

864 26 (1

2)(3

4)(5

6)(7

8), (2

3

5

7

9)(4

6

8) 940 2o

S

470 27 (1

2)(3

4), (2

3

5)(4

6

7

8

9) 2560 21279

A

1280 28 (1

2

3)(4

5)(6

7), (3

4

6

9

5

8

7) 3120 21559:

S

1560 29 (1

2)(3

4), (2

5

3

6

8

9)(4

7) 3144 21571

A

1572 30 (1

2)(3

4)(5

6)(7

8), (2

3)(4

5

7

6

8

9) 3304 21651:

S

1652 31 (1

2)(3

4)(5

6)(7

8), (2

3

5)(4

7

8

9

6) 3980 2o

S

1990 32 (1

2

3)(4

5)(6

7), (3

4

6

7

9

5

8) 4320 22159:

S

2160 33 (1

2

3)(4

5)(6

7), (2

3

4)(5

6

8

7

9) 5760 22879

A

2880 34 (1

2)(3

4)(5

6)(7

8), (2

3

4

5

7)(6

8

9) 5964 2o

S

2982 35 (1

2

3)(4

5)(6

7), (3

4

5

8

6

7

9) 6300 2o

S

3150 36 (1

2)(3

4)(5

6)(7

8), (2

3)(4

5)(6

7

9) 7452 2o

S

3726 37 (1

2)(3

4)(5

6)(7

8), (2

3)(4

5

7)(8

9) 9288 24643:

S

4644 38 (1

2

3)(4

5)(6

7), (3

4)(5

6

8)(7

9) 12960 26479:

S

6480

No. 2-generators of

M

11 degree

Aut

(

F

)

1

b

2

ab

3

a

10

b

2

a

5,

aba

7

b

2 66

A

332 2

b

2

ab

3

a

10

b

2

a

5,

ab

3

a

2

bab

96

A

482 3

a

,

b

288 2143:

S

144 4

a

,

a

2

ba

2

ba

4

b

2

a

7 768 2383

A

384 5

a

,

b

3

a

4

ba

6

b

2

a

8 792 2197:

S

1982 6

a

,

a

7

ba

6

ba

10

b

1296 2647:

S

648 7

a

,

a

7

bab

2

a

2 1380 2 o

S

690 8

a

,

aba

3

ba

4

b

1792 2447

A

4482

a

= (1

6

9

8

4

3

2

10

7

11

5),

b

= (3

7

8

11)(4

5

9

6).

No. 2-generators of

M

12 degree

Aut

(

F

)

1

a

4

ba

2

b

3

a

6

b

5,

ab

3

a

3

b

5

a

10

b

7 18

A

18 2

a

4

ba

2

b

3

a

6

b

5,

b

7

a

8

b

3

a

9

ba

22

S

22 3

a

4

ba

2

b

3

a

6

b

5,

a

5

b

7

a

3

b

5

a

10

ba

22

S

22 4

a

4

ba

2

b

3

a

6

b

5,

aba

2

b

6

a

6

ba

3 36

S

36 5

a

4

ba

2

b

3

a

6

b

5,

a

7

b

5

a

10

b

5

a

5

b

3 36

A

182 6

a

4

ba

2

b

3

a

6

b

5,

a

6

b

3

a

4

b

5

a

6

b

40

S

40 7

a

4

ba

2

b

3

a

6

b

5,

ba

7

b

3

a

8

b

3 48

A

3 162

S

3 8

a

4

ba

2

b

3

a

6

b

5,

a

2

ba

2

b

3

a

10

b

3 48

A

48

No. 2-generators of

M

12 (continued) degree

Aut

(

F

) 9

a

4

ba

2

b

3

a

6

b

5,

a

7

b

3

a

9

b

7

a

5

b

5 60

S

20 o

S

3 10

a

4

ba

2

b

3

a

6

b

5,

a

2

b

7

a

5

b

5

a

7

b

3

a

6 63

A

21 o

S

3 11

a

,

a

7

b

5

a

9

b

3

a

4 64

S

64 12

a

,

a

2

b

6

a

6

b

3 64 231

A

32 13

b

3

a

6

b

3

a

10

b

9,

a

2

ba

7

b

3

a

8

b

3 90

S

30 o

S

3 14

a

,

ab

7

a

9

ba

7

b

8

a

4 96 247

A

48 15

a

4

ba

2

b

3

a

6

b

5,

a

5

ba

10

b

5

a

5

b

5 117

A

39 o

S

3 16

a

4

ba

2

b

3

a

6

b

5,

b

3

a

6

b

5

ab

9

a

4 120

A

40 o

S

3 17

a

4

ba

2

b

3

a

6

b

5,

a

3

ba

7

b

8

a

10

b

9

a

9 120

A

3 402 2

S

3 18

a

,

a

2

b

7

a

3

b

5

a

10

b

2 160

S

160 19

a

4

ba

2

b

3

a

6

b

5,

ba

4

b

7

a

8

b

5

a

6 162

S

54 o

S

3 20

a

,

a

3

ba

3

b

4

a

6

b

7

a

8 180

S

180 21

a

,

a

3

b

8

a

6

b

9 180

S

180 22

a

4

ba

2

b

3

a

6

b

5,

a

4

ba

6

b

5

ab

192

A

64 o

S

3 23

a

4

ba

2

b

3

a

6

b

5,

ab

3

a

5

b

5

a

7

ba

9 198

A

3 662

S

3 24

a

4

ba

2

b

3

a

6

b

5,

ab

7

a

5

b

5

a

7

b

5

a

5 216

A

3 722

S

3 25

a

4

ba

2

b

3

a

6

b

5,

b

3

a

10

ba

4

b

7

a

10 264

A

88 o

S

3 26

a

4

ba

2

b

3

a

6

b

5,

ba

8

b

3

a

9

b

3

a

6 288

A

96 o

S

3 27

a

,

a

3

ba

6

b

5

a

8

b

3

a

8 360

A

360 28

a

,

b

360

A

360 29

a

,

ab

2

ab

3

a

9

b

7

a

5 396

A

396 30

a

,

a

3

b

8

a

10

ba

2

ba

7 396

A

396 31

a

,

ab

2

a

4

b

672 2335

A

336 32

a

,

a

6

ba

7

ba

10

ba

2 1080 2539:

S

540 33

a

,

a

9

b

3

a

4

ba

2 1776 2887

A

888 34

a

,

ab

6

a

5

b

9 2120 21059:

S

1060 35

a

,

a

2

ba

7

ba

6

b

2 2288 21143

A

1144 36

a

,

b

6

a

10

ba

4

b

3

a

2592 21295

A

1296 37

a

,

a

3

ba

3

b

7

a

5

b

9

a

2 2808 21403:

S

1404 38

a

,

ab

2

ab

9

a

6

b

5

a

3384 21691:

S

1692 39

a

,

b

6

a

6

b

5

ab

9

a

2 3400 21699:

S

1700 40

a

,

a

2

b

9

a

5

b

5

a

7

b

8 4400 22199

A

2200 41

a

,

b

4

ab

3

a

6

b

3

a

2 9728 22431

A

24322

a

= (2

9

3

12

4

8

11

6

7

10

5),

b

= (1

2)(3

4

10

9

5

6

8

7

12

11).

No. 2-generators of

M

22 degree

Aut

(

F

)

1

a

7

b

8

ab

5

a

6

b

10,

a

7

b

4

a

7

b

10

a

6

b

7 33

S

33 2

a

7

b

8

ab

5

a

6

b

10,

a

9

b

6

a

2

b

7

a

4

b

2 33

A

33 3

a

7

b

8

ab

5

a

6

b

10,

ba

5

b

6

a

9

b

7

a

6 33

S

33 4

a

7

b

8

ab

5

a

6

b

10,

b

7

ab

4

a

8

b

6

a

7 33

A

33

No. 2-generators of

M

22 (continued) degree

Aut

(

F

) 5

a

7

b

8

ab

5

a

6

b

10,

ab

9

a

5

b

10

a

2

b

42

A

212 6

a

,

b

7

a

7

b

2

a

10

b

5

a

10 48

A

3 162 2

S

3 7

a

7

b

8

ab

5

a

6

b

10,

a

2

b

2

a

2

b

5

a

7

b

8 48

S

48 8

a

7

b

8

ab

5

a

6

b

10,

a

9

b

5

a

10

b

3

a

5

b

7 48

S

48 9

a

2

b

8

a

6

b

3

a

7

b

8,

ab

3

a

2

b

10

ab

60

A

3 202

S

3 10

a

7

b

8

ab

5

a

6

b

10,

a

10

b

6

a

5

b

6

a

8 66

A

332 11

a

2

b

8

a

6

b

3

a

7

b

8,

a

3

b

8

a

8

b

9

ab

4 72

A

3 242 2

S

3 12

a

2

b

8

a

6

b

3

a

7

b

8,

a

3

b

5

a

4

b

7

a

10

b

6

a

5 84

A

3 142 2

S

y 32 13

a

7

b

8

ab

5

a

6

b

10,

a

3

b

4

a

7

b

6

a

6

b

7 84

S

422 14

a

,

b

90

S

90 15

a

,

b

6

a

2

b

10

a

5

b

8 180 2 o

S

90 16

a

,

a

5

b

8

a

2

b

3

a

9

b

7 198

A

3 662 2

S

y 3 17

a

,

ab

9

a

4

b

6

a

2

b

9

a

9 216

A

3 722

S

3 18

a

,

b

3

a

4

b

7

a

4

b

3 288

A

288 19

a

,

b

6

a

4

b

7

a

10

b

4

a

7 360

A

120 o

S

3 20

a

,

a

3

b

3

a

5

b

8

a

7

b

4 420

A

3 1402

S

3 21

a

,

a

2

b

9

a

9

b

2

a

7

b

6 432

A

432 22

a

,

ab

2

a

6

b

6

a

9

b

5

a

2 480 2239:

S

240 23

a

,

a

8

b

9

a

5 480

A

480 24

a

,

a

5

b

3

a

2

b

6

a

2

b

3 486

S

486 25

a

,

b

4

ab

8

a

4

b

4

a

3 504

A

3 842 2

S

32 26

a

,

ab

6

a

7

b

8

a

3 576 2287:

S

288 27

a

,

a

3

b

5

a

6

b

5

a

7 868

S

4342 28

a

,

a

5

b

3

a

7

b

5

a

6

b

900 2 o

S

450 29

a

,

a

7

b

9

a

6

b

7

ab

4 1056 2527

A

528 30

a

,

a

10

b

2

a

2

b

7

a

10

b

9 1080 2539

A

540 31

a

,

b

2

ab

10

a

6 1296 2647

A

648 32

a

,

a

7

b

2

a

9

b

6

a

4 1440 2719:

S

720 33

a

,

a

6

b

6

a

6

b

10

a

7

b

4 1512 2755

A

756 34

a

,

a

5

b

2

a

6

b

6

a

8

b

5

a

5 1800 2899

A

900 35

a

,

a

3

b

9

a

4

b

7

a

6

b

5 1920 2959

A

960 36

a

,

b

7

ab

8

a

8

b

6

a

10 2232 21115

A

1116 37

a

,

b

4

a

9

b

3

a

5

b

8

a

4 2232 21115

A

1116 38

a

,

b

10

a

7

b

3

a

9

b

2

a

2244 2 o

S

1122 39

a

,

ab

9

a

7

b

8

a

7 2244 2 o

S

1122 40

a

,

a

4

b

8

a

6

b

10

ab

3 2376 21187:

S

1188 41

a

,

ab

10

ab

8

ab

5 2664 21331:

S

1332 42

a

,

b

5

ab

5

a

8

b

4

a

4 2688 2671

A

6722 43

a

,

a

2

b

4

a

10

b

6

a

10

b

3 2784 21391:

S

1392 44

a

,

a

3

b

4

a

8

b

9 3036 2 o

S

1518 45

a

,

a

2

ba

9

b

4

a

10 3036 2 o

S

1518

No. 2-generators of M 22 (continued) degree Aut(F) 46 a, a 6 b 4 a 4 b 6 a 5 b 4 3036 2 oS 1518 47 a, b 3 a 7 b 9 a 6 b 4 3072 21535 A 1536 48 a, aba 10 b 2 a 4 b 4 a 9 3240 21619: S 1620 49 a, b 8 a 4 b 10 a 7 b 5 3960 21979 A 1980 50 a, a 8 b 9 ab 10 a 3960 2 1979: S 1980 51 a, a 8 b 2 a 5 b 9 a 9 b 7 3960 21979 A 1980 52 a, a 8 b 6 a 6 b 7 a 5 3960 21979: S 1980 53 a, b 8 a 9 b 4 a 6 b 8 a 4 4032 21007 A 10082 54 a, a 4 b 5 a 6 b 6 a 6 b 9 4032 22015 A 2016 55 a, a 6 b 9 a 6 b 5 ab 2 4092 2 oS 2046 56 a, a 5 b 8 a 2 b 2 4092 2 oS 2046 57 a, a 5 b 8 ab 3 a 7 b 8 6000 22999: S 3000 58 a, a 5 b 5 a 10 b 9 a 6624 2 3311 A 3312 59 a, a 2 b 3 a 5 b 7 a 10 b 7 6656 21663 A 16642 60 a, a 3 b 10 ab 7 a 9 b 10 6660 2 oS 3330 61 a, b 3 a 5 b 9 a 9 b 5 a 5 6660 2 oS 3330 62 a, a 7 b 3 a 10 b 7 a 2 b 9 6720 23359 A 3360 63 a, a 4 b 4 a 8 b 3 ab 4 6840 23419 A 3420 64 a, ab 9 a 2 ba 8 b 8 a 7 7680 23839: S 3840 65 a, a 2 b 10 a 7 b 8 a 4 7896 21973: S 19742 66 a, a 4 b 7 a 8 b 6 ab 4 9216 22303 A 23042 67 a, aba 5 b 8 a 8 b 7 9216 22303 A 23042 68 a, a 3 b 2 a 10 b 7 a 4 b 6 9744 22435 A 24362 69 a, a 5 b 7 a 4 b 9 a 2 b 7 10304 22575 A 25762 70 a, a 2 b 7 a 8 b 9 a 7 b 4 10920 22729: S 27302 71 a, b 2 a 7 b 4 a 6 b 8 a 8 10920 22729: S 27302 a= (1
5
12
21
6
18
4
2
19
14
17)(3
20
8
22
16
7
9
13
15
10
11), b= (1
13
18
15
7
14
4
21
3
12
22)(2
11
19
16
5
8
6
20
10
9
17).

Acknowledgments

The authors would like to express their thanks to Dr. Akihide Hanaki for his kind advice and encouragement.

References

1] Conway, J., Curtis, R., Norton, S., Parker, R. and Wilson, R. (1985). Atlas of Finite Groups. Oxford:Clarendon Press.

2] Evans, M. (1993). T-systems of certain nite simple groups. Math. Proc. Camb. Phil. Soc.113, 9{22.

3] Gilman, R. (1977). Finite quotients of the automorphism group of a free group. Can. J. Math. 29, 541{551.

4] Hall, P. (1936). The Eulerian functions of a group. Quart. J. Math. (Oxford Ser.)

7, 134{151.

5] Magnus, W., Karrass, A. and Solitar, D. (1976). Combinatorial Group Theory (Second Revised Edition). New York:Dover Publication.

6] Neumann, B. H. and Neumann, H (1951). Zwei Klassen charakteristischer Un-tergruppen und ihre Factorgruppen. Math. Nachr.4, 106{125.

7] Piccard, S. (1949). Sur les bases du Group Alterne. Comment. Math. Helvetici

23, 123{151.

8] Piccard, S. (1980). Diverses facons de caracteiser un groupe abstrait dont on connait une realisation. Publ. du Centre de recherches en Mathematiques pures, Neuch^atel, Serie I 15, 5{12.

9] Schert, M. et al. (1995). GAP - Groups, Algorithms and Programming. Lehrstuhl D f Mathematik, Rheinisch Westfalische Technische Hochschule, Aachen, Ger-many, fth edition.

10] Stork, D. (1972). The action of the automorphism group ofF

2upon the A

6 ;and PSL(2
7);dening subgroups ofF

2. Trans. Amer. Math. Soc.

172, 111{117.

11] Wielandt, H. (1964). Finite Permutation Groups. New York: Academic Press.

Osamu Higuchi

Sinku-Jouhou-Sisutemu Co. Iida 2-6-6 Kofu 400-0035 Japan Izumi Miyamoto

Department of Computer Science, Yamanashi University, Takeda 4-3-11 Kofu 400-8511 Japan